I think your co-worker is smart and you are also correct.
The useful things I found that putting everything into the headers is that:
No need for writing & sync headers and sources.
The structure is plain and no circular dependencies force the coder to make a "better" structure.
Portable, easy to embedded to a new project.
I do agree with the compiling time problem, but I think we should notice that:
The change of source file are very likely to change the header files which leads to the whole project be recompiled again.
Compiling speed is much faster than before. And if you have a project to be built with a long time and high frequency, it may indicates that your project design has flaws. Seperate the tasks into different projects and module can avoid this problem.
Lastly I just wanna support your co-worker, just in my personal view.